${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{n\le x}{y}^{\omega (n)}}=O_y(x{(logx)}^{y-1})$ for $y\ge 0$

Within this entry, $\omega$ refers to the number of distinct prime factors function, $\lfloor \cdot \rfloor$ refers to the floor function, $\log$ refers to the natural logarithm, $p$ refers to a prime, and $k$ and $n$ refer to positive integers.

Theorem 1.

For $y\mathrm{\ge }\mathrm{0}$, ${\displaystyle \mathrm{\sum }_{n\mathrm{\le }x}}{y}^{\omega \mathit{}\mathrm{(}n\mathrm{)}}\mathrm{=}{O}_y\mathit{}\mathrm{(}x\mathit{}{\mathrm{(}\log \mathit{}x\mathrm{)}}^{y\mathrm{-}\mathrm{1}}\mathrm{)}$.

Proof.

Since $y^{\omega (p^k)}=y$ for all $p$ and $k$, the real-valued nonnegative multiplicative function $y^{\omega (n)}$ the Wirsing condition with $c=y$ and $\lambda =1$. Thus:

${\displaystyle \sum _{n\le x}}{y}^{\omega (n)}$	$=O_y\left({\displaystyle \frac{x}{\log x}}{\displaystyle \sum _{n\le x}}{\displaystyle \frac{y^{\omega (n)}}{n}}\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}{\displaystyle \prod _{p\le x}}\left(1+{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{y^{\omega (p^k)}}{p^k}}\right)\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}\left(\exp \left({\displaystyle \sum _{p\le x}}{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{y}{p^k}}\right)\right)\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}\left(\exp \left(y{\displaystyle \sum _{p\le x}}{\displaystyle \sum _{k=1}^{\lfloor \frac{\log x}{\log p}\rfloor }}{\displaystyle \frac{1}{p^k}}\right)\right)\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}(\exp (y(\log (\log x)+O(1))))\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}(\exp (\log {(\log x)}^y))\right)$
	$=O_y\left({\displaystyle \frac{x}{\log x}}{(\log x)}^y\right)$
	$=O_y(x{(\log x)}^{y-1})$

∎

Title	${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{n\le x}{y}^{\omega (n)}}=O_y(x{(logx)}^{y-1})$ for $y\ge 0$
Canonical name	displaystylesumnleXYomeganOyxlogXy1ForYge0
Date of creation	2013-03-22 16:11:22
Last modified on	2013-03-22 16:11:22
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	9
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 11N37
Related topic	AsymptoticEstimate
Related topic	DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2
Related topic	WirsingCondition